Why does applying smoothness to a mesh reduce the number of vertices?
The shading options (Smooth and Flat) refer to how the vertex normals are calculated, which then essentially tells the renderer how to bounce light off edges.

For smooth shading, the vertex normals are calculated by averaging each face normal that they belong to. The face normal is simply the direction perpendicular to face, on the outward side.

For flat shading, each vertex can be considered to have multiple normals, one for each face it is part of, in the direction of the face. Many programs represent this by splitting each vertex into multiple vertices, one for each adjacent face. This is most likely what you are seeing in Unity, as that is most likely what the renderer in Unity is doing.

There is another kind of 'smoothing' called subdivision, which as its name suggests, splits faces into smaller faces. This obviously adds vertices, and is probably where you got the idea that smoothing adds vertices.